Related papers: Gradient estimates for Stokes and Navier-Stokes sy…
We consider a stochastic interacting particle system in a bounded domain with reflecting boundary, including creation of new particles on the boundary prescribed by a given source term. We show that such particle system approximates 2d…
Consider the unforced incompressible homogeneous Navier-Stokes equations on the $d$-torus $\mathbb{T}^d$ where $d\geq 4$ is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $u\in L^{2}(\mathbb{T}^d)$.…
We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsed\`a and Misiurewicz and contributing to a question of Navas on the…
Exponential stabilizability of the incompressible Navier-Stokes equations under dynamic slip boundary conditions toward arbitrary time-dependent trajectories is proven. The feedback control law is constructed explicitly using oblique…
We prove short-time well-posedness and existence of global weak solutions of the Beris--Edwards model for nematic liquid crystals in the case of a bounded domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The system…
We prove a sufficient conditions of local regularity of suitable weak solutions to the MHD system for the point from $C^3$-smooth part of the boundary. Our conditions are the generalizing of the Caffarelli--Kohn--Nirenberg theorem for…
We prove convergence of a finite difference approximation of the compressible Navier--Stokes system towards the strong solution in $R^d,$ $d=2,3,$ for the adiabatic coefficient $\gamma>1$. Employing the relative energy functional, we find a…
In this paper, we establish the global existence of small solutions to the inhomogeneous Navier-Stokes system in the half-space. The initial density only has to be bounded and close enough to a positive constant, and the initial velocity…
In this paper, we mainly investigate the Cauchy problem for the incompressible Navier-Stokes equations in homogeneous Besov spaces $\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq p<\infty,\ 1\leq r\leq \infty, \ d\geq 2$. Firstly, we prove the…
It is well known that the solution of the 3d Navier--Stokes equations remains bounded if the initial data and the forcing are sufficiently small relative to the viscosity, and for a finite time given any bounded initial data. In this…
In this paper, we consider weak solutions of the Euler-Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling…
We investigate uniqueness of weak solutions for a system of partial differential equations capturing behavior of magnetoelastic materials. This system couples the Navier-Stokes equations with evolutionary equations for the deformation…
The paper is aimed at analysing a singular perturbation of the Navier-Stokes equations on a compact closed manifold. The case of compact smooth manifolds with boundary under the Dirichlet conditions is also included. Global existence and…
We consider the homogenization of a Poisson problem or a Stokes system in a randomly punctured domain with Dirichlet boundary conditions. We assume that the holes are spherical and have random centres and radii. We impose that the average…
In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is…
In this article, a perturbation theory of the compressible Navier-Stokes equations in $\mathbb{R}^n$ $(n \geq 3)$ is studied to investigate decay estimate of solutions around a non-constant state. As a concrete problem, stability is…
We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in $\mathbb{R}^d, \ d\geq 2$. A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero.…
Systems of hydrodynamic type equations derived from the Navier-Stokes equations and the boundary layer equations are considered. A transformation of the Crocco type reducing the equation order for the longitudinal velocity component is…
On a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \geq 3$, we continue the study of Shen and of Kunstmann and Weis of the Stokes operator on $\mathrm{L}^p_{\sigma} (\Omega)$. We employ their results in order to determine the…
This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (non-constant density and viscosity) incompressible Navier-Stokes system on a bounded domain. The proposed method…